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Rational Functions Analysis and Graphing Our Learning objective: Is to explore and explain why the denominator of a rational function cannot be zero. Thus recognizing these values as the places where vertical asymptotes occur, (which are disastrous things to have), and graphically what vertical asymptotes look like and mean. What is a Rational Function? • A rational function has the form of a polynomial over a polynomial. • The bottom polynomial must never be zero! • If the bottom polynomial is zero this will make the function undefined. • Hence, these values are left out of the function’s domain. Equations of rational functions Self Check 1 Which function below is a rational function? A.) B.) Answer What are Vertical Asymptotes? • Vertical asymptotes are the values that make the denominator go to zero, which makes the function undefined. • These values are represented by x = a and/or x = b, where a and b are real numbers. • What do you notice about the equations x = a and x = b? • They are the equations of vertical lines, hence the name vertical asymptotes! Basic Rational Function Shape. How do we find Vertical Asymptotes? • We have to find the places where the denominator goes to zero. • We do this by setting the denominator (bottom polynomial) equal to zero and finding the values of x that make it zero. • This is when we get the equations of the form x = a. Vertical line at x = a. Self Check 2 • What are the vertical asymptotes for the function: • Answer Do Vertical Asymptotes Always Exist? • No. If the zeros of the bottom polynomial are complex numbers, then the function does not have vertical asymptotes. • See the diagram to the right? This rational function’s denominator does not go to zero. We can tell because the ends of the graph get close to the x – axis but do not cross the x- axis. Self Check 3 • Which of the following functions has vertical asymptotes? Set each denominator equal to zero and solve for the values of x. A.) B.) Answer How can we Identify Vertical Asymptotes Graphically? • Vertical asymptotes are identified as dashed or dotted vertical lines in the plane. • You guessed it!! The equations of those vertical lines are the values of x that make the denominator equal to zero or x = a and x = b. What do you notice about the graph of the function as it approaches the vertical line that is the vertical asymptote? Self Check 4 Which diagram illustrates a vertical asymptote? A.) B.) Answer What do Vertical Asymptotes mean to the graph of the function? • Since the vertical asymptotes make the function undefined, the graph of the function NEVER crosses or touches the vertical asymptote. • Hence, the graph bends and diverges to positive infinity or negative infinity on each side of each vertical asymptote! • See the next slide for two examples!! Graphs of Functions Containing Vertical Asymptotes. Rational Function with one This rational function has vertical asymptote. It two vertical asymptotes. So diverges, bends toward you have to determine positive infinity on the right which way it diverges on and bends toward negative each side of each vertical infinity on the left. asymptote. Here that is four different calculations!! What is the most vertical asymptotes a function can have? The tangent function has infinite!! Self Check 5 Tell which way the function diverges as it approaches the vertical asymptote from the right and the left. Answer Are Vertical Asymptotes Good? • What do you think it means to say the graph bends or diverges toward positive or negative infinity? • If you are analyzing the cost of producing something and the model’s graph bends toward positive infinity, do you think that is something good? • Vertical asymptotes are disastrous things, because when the function diverges it means the function forever goes in the direction of infinity. Thus we call the function undefined. • We avoid models that behave in this way because they are unstable and can have disastrous effects. Self Check Answers 1. A. 2. There are two: x = - 1/2 and x = 1/2 3. B. 4. B. 5. From the left goes toward negative infinity and from the right toward positive infinity.

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